Radiative transfer in a cylinder. II. Special problems. Asymptotics

The method of computing the radiation field in an infinite circular cylinder proposed in Part I is now applied to the case of isotropic scattering with sources on the boundary and axis of the cylinder, as well as for a uniform distribution of sources inside the cylinder. For the simplest kernel we obtain exact solutions of the basic integral equation in explicit form. For scattering in a spectral line with complete frequency redistribution and a power absorption profile we develop an asymptotic theory for the case when the optical radius of the cylinder is large. We solve the asymptotic equations for the basic characteristics of the scattering in closed form for conservative scattering and find its asymptotics. We obtain estimates of the mean number of scatterings with a layered source, and also the mean and variance of the number of scatterings with a uniform source distribution.Translated fromAstrofizika, Vol. 37, No. 4, 1994.This work was carried out with the financial support of the Russian Basic Research Fund (grant 93-02-2957).     

Effects of Thomson scattering on the shape of a spectral line

The effect of Thomson scattering and Compton effect on the shape of a spectral line is studied in the case of a plane-parallel atmosphere. The theoretical values of red shift and intensity at the top of the line, having an equivalent width 5√π10^?10 cm, and of the red asymmetry are tabulated for 62 different values of the optical thickness τ′; from τ′=0.2/√3 to τ′=2.6/√3. The case of solar atmosphere is considered and it is shown that all main features of the observed red shifts can be interpreted by means of a scattering theory, with an error smaller than one milliängström. The need of further measurements is pointed out.     

Photometric properties of the surface of Io and their influence on line formation in the atmosphere

We give a quantitative theory of line formation in an atmosphere above a surface with backscattering properties. Sufficiently high spatial and spectral resolution spectra of resonance lines in Io region A can yield data on the surface scattering properties as well as the number density of scattering molecules. We discuss macroscopically homogenous models of scattering from the surface of Io and conclude that multiple reflection from crystal facets is the most likely cause for the observed geometric albedo and phase variation.     

Modulation of galactic cosmic ray anisotropy in heliomagnetosphere: Influence of cosmic ray scattering on sidereal daily variation

In previous papers, the present authors have shown that the galactic anisotropy is modulated due to cosmic ray orbital deflection in the heliomagnetosphere, and that the sidereal time daily variations of galactic origin can be expressed using the basic vectors, which have been obtained by calculating trajectories of cosmic rays in a model magnetosphere having Parker's Archimedian spiral structure with a flat or a wavy neutral sheet. In the present paper, the magnetic irregularities superposed on the Parker's spiral field have been taken into account, which cause the scattering of cosmic rays and disturb their orbits. We examined the fluctuations of asymptotic directions calculating their orbits by the Monte-Carlo simulation, based on the theory of the multiple scattering process. It is shown that the dispersion of the projected deviation angle is determined mainly by the scattering mean free path and by the structure of the order magnetic field, e.g. the polarity state of the heliomagnetosphere and the extent of the neutral sheet. We investigated also the influence of the fluctuations of asymptotic directions on the sidereal daily variation. It is found that, under some conditions, the scattering causes only the attenuation of the amplitude of the basic vector, and does not change its phase. The attenuation is negligibly small at high rigidities larger than ~ 1000 GV, but becomes more serious with decreasing rigidity. The rigidity dependence curve of the attenuation rate was calculated for various cases. A simple and approximate method is also presented for the derivation of those curves for any value of the magnitude of the mean free path and for various model magnetospheres. It is noted, however, that the lower limiting rigidity below which the present method is not applicable is relatively high in the Positive polarity state.     

The Voigt profile of the decameter radio recombination lines and electron density of the expanding CII region towards Cassiopeia A

The carbon decameter radio recombination line (RRL) shape is described by a Voigt profile, since explicit line broadening is observed in the wings. A Lorentz component line half width is determined by the method of a Lorentz and Gauss line profile fitting of a curve. Since the Lorentz line shape is described by the Stark effect and the interaction with the Galaxy non-thermal background radiation, then by comparing the Lorentz component experimental and theoretical values a medium component electron density can be calculated for an expanding CII region towards Cassiopeia A. So far as the decameter lines are formed due to the transitions between the levels with large principal quantum numbers n>600, the life time, and thus the line width are very large. The collision transition rate is described by the perturbation theory for a low temperature medium. The radius matrix elements for a highly excited atom are defined more exactly with determining the transition rates for radiation interaction and scattering.     

Generalized compton effect and resonance scattering in astrophysical spectra

The analysis of the Skylab measurements on the ultraviolet limb spectra confirm the presence of a generalized Compton effect in the solar spectrum which can be explained by Thomson scattering theory. The present measurements on the Orion spectrum and interstellar line towards -Arae give a large generalized Compton effect which could be explained by a resonance scattering theory. These numerical results cannot be due to random errors in the measurements, as follows from the statistical discussion. The need for further measurements is pointed out.     

Multiple light scattering: mean number of scatterings and related problems

This is a discussion of V. A. Ambartsumyan’s studies of the mean number of scatterings for photons in scattering media and of further work and development in this area, especially at Ambartsumyan’s St. Petersburg school. The following questions are discussed briefly: (a) the traditional method for calculating the number of scatterings from the source function and critiques of this method. (b) The equation for the number N(τ; τ₀ ) of scatterings for a photon born at optical depth τ in a plane layer of optical thickness τ₀ and its use for calculating the number of scatterings, averaged over the entire ensemble of photons for a medium with arbitrary internal sources. These questions are first considered for the case of monochromatic scattering, and then for scattering in a spectral line with complete frequency redistribution (CFR). (c) The mean path length for a resonance line photon in a scattering medium with CFR and continuum absorption: the basic equations and asymptotic behavior of an optically thick layer. (d) A review of calculations of and in media that are so thick that the CFR approximation breaks down and the effects of partial frequency redistribution (PFR) become dominant. The presentation is at a semiquantitative level in many parts of this paper, with stress on physical significance rather than the mathematics, through the use of approximate and asymptotic solutions. Translated from Astrofizika, Vol. 52, No. 1, pp. 29–45 (February 2009).     

Transfer of resonance radiation and photon random walks

Transfer of resonance radiation in an infinite medium is considered as a process of random walks of photons. Close relation is shown to exist between the problems of transfer of line radiation and the stable distributions of the probability theory. This relation is used as a basis of a new method for the investigation of the asymptotic properties of the radiation field far from the sources.     

Astrophysical spectroscopy and neutron reactions: Integral transforms and Voigt functions

The Voigt functionsK(x, y) andL(x, y) which play an essential role in astrophysical spectroscopy and neutron physics are investigated and generalized from the viewpoint of integral operators. Unified representations and series expansions involving classical functions of mathematical physics and multivariable hypergeometric functions are established. From the delicate asymptotic analysis of Laplace and Hankel integral transforms we extract complete and rigorous asymptotic expansions of the generalized Voigt functions for large values of the variablesx andy which are of great value in the theory of spectral line profiles.     

Quasi-linear perpendicular diffusion coefficients of charged cosmic rays calculated directly from the Newton–Lorentz equation

It is the subject of this article to determine diffusion coefficients of charged test particles perpendicular to a mean magnetic field analytically by starting directly from the Newton–Lorentz equation. We employ the so-called standard quasi-linear theory to test the well-known field line random walk limit. It is demonstrated in this article that there is an additional gyroresonant contribution which describes the scattering of the particle away from a single field line. For certain situations, this additional contribution is small and can be neglected. For other cases, we obtain new results such as a rigidity-dependent quasi-linear perpendicular mean free path. The results of this article are relevant for particles in the solar wind which move scatter free or nearly scatter free in the direction parallel to the magnetic field of the Sun.     

A Monte Carlo study of polarization structures in the Thomson-scattered line radiation

Thomson scattering is often invoked to explain broad wing features that are seen in various objects including active galactic nuclei and symbiotic stars. Despite the wavelength-independent scattering cross-section of Thomson scattering, the line flux may exhibit wavelength-dependent linear degree of polarization, because various parts of emission wings are contributed by photons with different scattering numbers. Specifically, more scattered and hence more weakly polarized photons tend to fill the farther wing parts from the line centre, while the neighbourhood of the line centre is dominated by less-scattered photons with higher degree of polarization. Using a Monte Carlo technique, we investigate the polarization structure of Thomson-scattered line radiation. A detailed analysis of polarization structure formation is conducted by investigating the dependence of the polarization and profile width on the scattering number for various finite electron scattering slabs. Significantly varying degree of polarization is obtained when the scattering medium has Thomson optical depth  τ_Th≥ 1  . We present our high-resolution spectrum of the symbiotic star V1016 Cyg obtained with the Bohyunsan Optical Echelle Spectrograph (BOES) in order to fit the broad profile around Hα by electron scattering wings adopting an oblate spheroidal geometry with Thomson optical depth  τ_Th= 0.5  and electron temperature   T _e= 6.2 × 10⁴  K  . Local maxima in the linear degree of polarization of Thomson-scattered line radiation are expected to appear in the spectral regions characterized by the average scattering number ≃1.     

Radiation transfer in a cylinder. III. Spectrum of basic integral equation

We present a method for computing the spectrum of the integral equation for radiation transfer in a cylinder. This method, as in the previous articles in this series, is based on a Hankel transformation applied to the equation. Calculating the spectrum then reduces to solving the equation for the auxiliary function for each eigenvalue separately. The corresponding eigenfunction is then found by an additional integration. We find asymptotic expressions for the eigenvalues and the eigenfunctions for a cylinder with a large optical radius when there is scattering in a spectral line, with complete redistribution over frequency when the absorption coefficient obeys a power law. We also derive equations determining the quantities entering into these expressions. For the simplest kernel of the equation all quantities can be expressed in terms of Bessel functions and roots of a transcendental equation.Translated from Astrofizika, Vol. 38, No. 1, pp. 75–88, January–March, 1995.     

Angular fluctuations in the cosmic microwave background owing to initial perturbations in the radiation temperature

A new method is proposed to account for multiple scattering by electrons in calculations of the correlation functions describing the angular fluctuations in the cosmic microwave background radiation (CMBR). The apparatus of the theory of radiative transport with Rayleigh scattering is used. The problem is reduced to solving an integral equation for the vector source function (dependent only on time), along with differential equations for the other quantities (scalar potentials, baryon velocities, etc.) which show up in the problem. The quantities which describe the angular fluctuations in the CMBR (in the temperature and in the polarization) are then calculated by integrating the vector source function along the line of sight. As an illustration, the correlation functions and power spectra are calculated for the case where the fluctuations are produced by some initial gaussian perturbations of the CMBR. __________ Translated from Astrofizika, Vol. 50, No. 4, pp. 621–631 (November 2007).     

行星际闪烁(IPS)单站观测对太阳风速度的诊断

借助于弱散射理论和模式拟合方法，单站行星际闪烁观测可以诊断太阳风速度，本文讨论了太阳风参数和射电源角尺度对闪烁谱的影响，以及太阳风速度的积分效应，结果表明，闪烁谱的特征是与视线上距太阳最近处的太阳风速度直接相关的。     

Effect of continuum absorption on the polarization of resonance lines

Multiple resonance scattering of radiation in a spectrum line with absorption in the continuum is examined. It is assumed that the scattering atmosphere is semi-infinite and that there is no magnetic field or continuum emission at the frequencies of the spectrum line. The polarization characteristics of the emerging radiation are determined for unpolarized primary radiation sources distributed uniformly within an atmosphere in the case of a Voigt absorption profile. The calculations employ an iterative solution of the Ambartsumyan-Chandrasekhar matrix integral equation.     

Multiple Scattering of Light in Regolith-like Media: Geometric Optics Approximation

We present a numerical model that allows us to calculate the contribution of a specified scattering order in the geometric optics approximation for media composed of large particles with an arbitrary phase function. It has been demonstrated that the correlated propagation of the incident and emergent rays, which is disregarded in the classical radiative-transfer theory, markedly affects the contributions of different orders of scattering, especially the first-order scattering. If the theory describing the photometric properties of regolith-like media ignores the shadowing effect, the errors of its application may reach several tens of percent even for bright surfaces. The packing density of a medium essentially influences the phase dependence of the first-order scattering, although its effect on the value and the phase curve of the higher scattering orders is relatively weak. The backscattering peaks calculated on the basis of the Hapke theory are narrower than those obtained from the numerical simulations, because the Hapke theory is an approximate approach.     

Partial frequency redistribution of polarized radiation

Resonance polarization, which is created by the scattering of an anisotropic radiation field in regions of zero or weak magnetic fields, is strongly dependent on the frequency redistribution taking place during the scatterings. Here we discuss the frequency redistribution matrix relevant to resonance lines, concentrating on linear polarization. First we analyze in detail the redistribution matrix in a zero magnetic field given by the theory of Omont, Smith and Cooper (1972), revisited by Domke and Hubeny (1988). We explain that the linear polarization maxima which may appear in the wings of the Stokes Q profiles of strong resonances lines such as the Ca I 4227 Å line are coherent frequency redistribution effects. Various approximate forms of the frequency redistribution matrix are also examined. For resonance polarization in a weak magnetic field, we suggest a new expression for the redistribution matrix which can be used at all line frequencies, and is consistent with the condition that the Hanle effect acts only in the line core.     

On the influence of resonant scattering on cosmic microwave background polarization anisotropies

We implement the theory of resonant scattering in the context of cosmic microwave background (CMB) polarization anisotropies. We compute the changes in the E-mode polarization (EE) and temperature E-mode (TE) CMB power spectra introduced by the scattering on a resonant transition with a given optical depth τ_X and polarization coefficient E ₁. The latter parameter, accounting for how anisotropic the scattering is, depends on the exchange of angular momentum in the transition, enabling observational discrimination between different resonances. We use this formalism in two different scenarios: cosmological recombination and cosmological re-ionization. In the context of cosmological recombination, we compute predictions in frequency and multipole space for the change in the TE and EE power spectra introduced by scattering on the Hα and Pα lines of hydrogen. This constitutes a fundamental test of the standard model of recombination, and the sensitivity it requires is comparable to that needed in measuring the primordial CMB B-mode polarization component. In the context of re-ionization, we study the scattering off metals and ions produced by the first stars, and find that polarization anisotropies, apart from providing a consistency test for intensity measurements, give some insight on how re-ionization evolved. Since polarization anisotropies have memory of how anisotropic the line scattering is, they should be able to discern the O  i 63.2-μm transition from other possible transitions associated to O  iii , N  ii , N  iii , etc. The amplitude of these signals are, however, between 10 and 100 times below the (already challenging) level of CMB B-mode polarization anisotropies.     

Linear fREDHOLM integral equations for radiative transfer problems in finite plane parallel media. I. Imbedding in an infinite medium

Linear FREDHOLM integral equations are derived for the STOKES vector of the radiation emerging from a scattering plane parallel medium of finite optical thickness. The integral equations are obtained by means of imbedding the slab in an infinite medium. They are formulated in terms of GREEN 's function matrices and renormalized for the asymptotic eigenmode. Explicitly, linear integral equations are given for the reflection and transmission matrices. The reciprocity principle is employed to obtain integral equations also for the mean intensity of the inner radiation field in the case of the slab albedo problem.     

High-resolution Martian atmosphere modeling

A multilayer radiative transfer, high-spectral-resolution infrared model of the lower atmosphere of Mars has been constructed to assess the effect of scattering on line profiles. The model takes into account aerosol scattering and absorption and includes a line-by-line treatment of scattering and absorption by CO₂ and H₂O. The aerosol complex indices of refraction used were those measured on montmorillonite and basalt chosen on the basis of Mars ir data from the NASA Lear Airborne Observatory. The particle sizes and distribution were estimated using Viking data. The molecular line treatment employs the AFGL line parameters and Voigt profiles. The modling results indicate that the line profiles are only slightly affected by normal aerosol scattering and absorption, but the effect could be appreciable for heavy loading. The technique described permits a quantitative approach to assessing and correcting for the effect of aerosols on lineshapes in planetary atmospheres.